The AKS Sorting Network

Abstract

In 1983, M. Ajtai, J. Komlos, and E. Szemeredi described a sorting network that requires C * log n steps to sort n keys where C is a large constant—scholars usually refer to these sorting networks as the AKS networks [Ajtai, Komlos, Szemeredi (1983) Combinatorica 3:1–1; Ajtai, Komlos, Szemeredi (1983) Proceedings of the ACM symposium on theory of computing pp 1–9.]. The AKS sorting network has O(log n) layers with a complete binary tree on each layer with O(n) nodes. Each node of each tree contains one (λ, σ, ε)-separator circuit. The separator in each node will tend to move low strangers and high strangers to their designated layers. The keys circulate in the different layers of the tree until they eventually collect in the last layer which will complete the sorting. No one seems to know the exact value of the complexity constant(C) of the AKS sorting algorithm. But many believe it is in the hundreds at least. If we assume that C = 87, then an AKS network is faster than a merge-sorting network only when the number of keys is greater than 1.2 * 10⁵². Thus, the ASK sorting network, despite being theoretically optimal, is impractical. The execution time of many parallel algorithms grows logarithmically or poly-logarithmically. Thus, complexity constants become important here.